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STATO O MERCATO? Intervento pubblico e architettura dei mercati

Pavia, Università, 5 - 6 ottobre 2001

FRANCESCA BARIGOZZI - BERTRAND VILLENEUVE

INFLUENCING THE MISINFORMED MISBEHAVER: AN ANALYSIS OF PUBLIC POLICY TOWARDS

UNCERTAINTY AND EXTERNAL EFFECTS

pubblicazione internet realizzata con contributo della

Società italiana di economia pubblica

Dipartimento di economia pubblica e territoriale – Università di Pavia

In‡uencing the Misinformed Misbehaver:An Analysis of Public Policy towards

Uncertainty and External E¤ects¤

Francesca Barigozziy Bertrand Villeneuvez

May 2001

Abstract

We study a situation in which government in‡uences consumers’ be-haviors by providing both information and incentives. More generally, wepropose a methodology for solving models of signal cum cheap talk.

We develop the case of consumption choice in the presence of uncer-tainty and external e¤ects. The instruments used by the government areinformation campaigns and taxes. A di¢culty arises because the gov-ernment would like to bolster these less than perfectly e¤ective tools bydelivering biased information to the misbehaver. We study the equilib-rium trade-o¤ between informing and o¤ering incentives. Environmentaltax policy, anti-smoking campaigns and policy against over-consumptionof antibiotics serve as illustrations.

Keywords: information campaigns, tax policy, cheap talk, signaling,skeleton.

JEL: I18, H30, D82.

1 IntroductionAs for the “mad cow” disease and other hotly debated issues concerning publichealth, food safety and the environment, risk controversies have mushroomed.

¤We have received valuable comments, suggestions or corrections from Giacomo Calzolari,Helmuth Cremer, Thomas de Garidel, Bruno Jullien, Alessandro Lizzeri, Maurice Marchandand seminar participants in Bologna, Bolzano, Madrid, Paris, Seattle. Hélène Bories, PatrickCourvalin and Anna Mingardi helped us with the account of antibiotics in the introduction.Errors, in any case, are our own exclusive responsibility.

yUniversity of Bologna.zCEA and IDEI, University of Toulouse. Corresponding author: Bertrand Villeneuve,

IDEI, Université de Toulouse 1, 21 allée de Brienne, 31000 Toulouse, FRANCE. E-mail:bertrand.villeneuve@cict.fr.

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Since policy makers must often assess and communicate such risks, the individ-uals’ con…dence in government or other authorities is a decisive component ofpolicy-making. Our work focuses on communication as disclosing the con‡ictsbetween a benevolent authority and consumers. Two ingredients are indispens-able: the public’s ignorance of the risk to be regulated and the impossibility forindividuals alone to rightly internalize certain negative consequences of their ac-tions. In isolation, each is relatively easy to solve: the former needs informationprovided to the public; the latter, optimized incentives. But in combination theseremedies interfere with one another and result in political confusion when incen-tives and coercive instruments are defective. The maximization of social welfaredoes not necessarily imply truthful policies, and consumers know it. Indeed, theyare increasingly skeptical of promotional strategies that use the outcome of thescienti…c literature or cite expert advisory committees.

In the present paper, the policy-making process is analyzed as a game in whichgovernment wants to in‡uence consumers’ behaviors through both tax policy andinformation campaigns, and where rational consumers react in a Bayesian man-ner. Instruments being imperfect, the government is often tempted to “improve”behavior by providing biased information. Con…dence, we show, is not easilycontrolled. Depending on the coordination between government and consumers,the same background data can produce various policies and real e¤ects. We de-termine the structure of practicable policies and discuss the trade-o¤ betweenvagueness in communication and distortion of incentives.

In‡uence Games We introduce a general methodology for tackling in‡uencegames, i.e. games in which the principal is an informed party who combinesdi¤erent instruments to transmit information and provide incentives to the agent.The relevant literature was initiated by Crawford and Sobel (1982) and Milgromand Roberts (1986). Crawford and Sobel (1982) show that the precision of theinformation depends on the intensity of the con‡ict between the two parties’objectives. Quite recently, these ideas have been applied to political games inwhich a lobby tries to in‡uence policy makers.1 In a major contribution tothe rational foundations of advertising, Milgrom and Roberts (1986) model a…rm that signals its product quality through price and dissipative advertising(burned money) to enhance consumers’ willingness to pay for the product. Inline with these articles, and contrary to Maskin and Tirole (1992), we are notmainly interested in characterizing optimal mechanisms; rather, we study thecombination of imperfect mechanisms, giving priority to their practical structureand consequences.

In a recent development along these lines, starting from the classical model ofCrawford and Sobel, Austen-Smith and Banks (2000) show how burning money

1See Helpman (2000), Presidential Address at the Econometric Society World Congress,Seattle and Grossman and Helpman (2001), for surveys.

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can improve cheap talk. In particular, they clearly show why the informationtransmitted can be perfect, and why the most informative equilibrium need notbe the most e¢cient.

We retrieve these results in our context but using a distinctively di¤erentmethodology, that is conceived to be applicable to a variety of models thus en-abling us to make other useful …ndings. First we change the perspective: we showthat cheap talk is almost useless when costly signals are available2 and explainwhy more precise equilibria are typically more distorted. Second, rather thansearching for the equilibria of a given economy, we de…ne the minimal amount ofinformation (a skeleton) to describe an equilibrium and determine the full set ofeconomies that admit a given skeleton as equilibrium. We prove the uniquenessof the fully revealing equilibrium and characterize it in detail. More generally,the skeleton approach helps understanding of the interaction between costly andfree signals and opens the way to interesting comparative statics.

The Analysis of Health and Environmental Policy Our model deals withpolicies that a¤ect the consumption of commodities that are detrimental to con-sumers’ welfare both individually and collectively. Typically, side e¤ects are dueto individual consumption; external e¤ects are due to overall consumption inthe economy. Broad-spectrum antibiotics display (apart from the obvious bene-…ts) this double negative impact. At the individual level they clear the way toopportunistic infection by more resistant germs;3 and at the societal level, theyenhance the resistance of the germs involved in contagious diseases. Analogously,in the case of tobacco and alcohol, one can readily distinguish between diseaserelated to individual consumption and the social damage from passive smoking orthe cost to the health care system (not to mention psychosocial issues like drunkdriving and addiction).

These two types of negative e¤ects explain why, without government inter-vention in the form of information or incentives, consumers may not consumee¢ciently. First, side e¤ects are not necessarily recognized by consumers. Forexample, the real magnitude of side e¤ects of antibiotics is no more than a vaguenotion for most people; likewise, the risk smokers perceive can be under- orover-estimated (Viscusi 1990). Second, external e¤ects (e.g. the rise of resistantstrains, passive smoke) are very largely ignored by consumers in the absence ofincentives such as taxes, norms, controls.

2Our proof is direct; an indirect proof is given in Manelli (1996).3Some broad-spectrum antibiotics decrease the individual’s immunological response, and

as a consequence new diseases can arise. For example, many antibiotics based on penicillinare used to treat diseases like bronchitis, otitis and tonsillitis caused by di¤erent bacteria(staphilococcus aureus, haemo…lus in‡uenzae, streptococcus pneumoniae). Possible side e¤ectsof penicillin consumption are candida albicans and herpes. See Levy (1992) for the medicalviewpoint and Brown and Layton (1996) for an excellent economic analysis of the externale¤ects.

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Political attitudes towards tobacco are typical of the schizophrenia we discussin the present work. E¢cient taxation is, generally, di¢cult to establish, butcompared to others levies, tobacco taxes are an easy source of funds. Govern-ment might try to optimize health and budgetary objectives by manipulatingconsumers’ beliefs on the individual consequences of smoking. Obviously, ratio-nal consumers form their opinion with this danger in mind, so the success ofsuch attempts is uncertain. In the same vein, let us remark that the pervasiveopinion that consuming a lot of antibiotics may cause individual resistance to thetreatment is unfounded. (Actually the problem is the resistance acquired by thegerms, which concerns the society rather than the individual strictly speaking).Authorities (which may not be directly responsible for this belief) are clearlytempted not to bother correcting it, since it serves (at a low cost) the practicalgoal of curbing consumption.

We assume that the government is better informed and benevolent and thatit maximizes the utility of the representative consumer.

To support our assumption that the government is better informed, note thatthe government may appeal to experts (civil servants, professionals, academics)who are able to transform dispersed data and results into operational knowledge.We do not need to assume that this operation is perfect, only that it is performedbetter by the experts than by the general public. Moreover, informing the pub-lic is a never-ending task. We know that discouraging teenagers from smokingrequire renovated strategies year after year. Though one may think that the “so-ciety” is already saturated with information on the relationship between tobaccoand cancer, each new cohort of consumers still has to be educated.4

Nevertheless, the government confronts the following dilemma: taxes are im-perfect instruments, and it is tempting to make them work better by disseminat-ing biased information. This engenders a sort of paternalism: the governmentwants consumers to consume e¢ciently, but, being unable to commit to neu-tral and truthful information transmission, it may send interested messages. Inour view, in all circumstances where there is some imperfection that impedeseconomic e¢ciency, taxes must be understood as a metaphor for the entire com-prehensive policy package, also comprising contracts, restrictions, standards andnorms.5

One crucial aspect of the model is the analysis of the tax as a signal thattransmits information on the value of side e¤ects. The tax has two consequences:…rst, by modifying the price it provides an incentive to internalize the external

4Publication of rough scienti…c …ndings in the mass media, though in principle a contributionto the formation of public opinion, may generate in confusion. Sensationalism and caricaturesare common. The most telling examples are extreme dietary recommendations.

5Causes of imperfection are often related to asymmetric information. In moral hazardmodels, for example, outcomes are observable, but the contribution of e¤ort cannot be perfectlyseparated from random e¤ects. As a consequence, contracts are rarely able to implement …rst-best e¢ciency.

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e¤ects; second, it acts as a signal of the value of side e¤ects.Information campaigns are analyzed as messages, i.e. short statements aiming

at informing individuals about the e¤ects of certain goods. The best example arewarning labels on cigarette packs or on the hazards of alcoholic beverage. Afundamental characteristic of these campaigns is that they have no direct impacton government’s or consumers’ utility: they are inexpensive, and to simplify, wecan class them as cheap talk messages.6 ;7 This implies that the literal meaning ofthe information conveyed is vague enough not to be falsi…able. Take the warninglabel “Seriously Harmful to Health” on cigarette packs. This is not false, but theexact nuance it carries is a matter of social convention (speci…cally, how cheaptalk is interpreted). Like taxes, information campaigns are imperfect instrumentsof policy.

Another crucial element is the analysis of tax distortions. In the model, thesign of the marginal cost of public funds is not restricted a priori. For this reason,the paper draws some practical conclusion from the literature on the “doubledividend”. In this literature, a tax on a polluting good is welfare-improving fortwo reasons: it reduces pollution and it also reduces the distortions caused bypreexisting taxes. Now it has been shown that the double dividend exists onlyunder certain conditions; in particular, this means that the sign of the marginalcost of public funds is not determined a priori (see Goulder 1995 for a survey).8

Finally we show, given benevolent authorities and rational consumers, thatthe major cause of trouble is lack of government credibility. By de…nition, then,government able to commit ex ante to inform truthfully would not encounter thedi¢culties we discuss. If such a government is in place, …ne. Yet a distrustfulattitude on the part of the public towards informed authorities is frequent: peopleoften feel that the government’s actions are motivated by economic interest morethat by the public interest (think of the di¤usion of information about the HIV-contaminated blood in the eighties in France and Germany, or “mad cow” diseasein Europe).

To solve the social game sketched here, we take an approach based on Bayesianequilibrium: people are not systematically fooled and the government tries tomake the best of the instruments available. We analyze one possible cause ofconsumers’ distrust, establishing the trade-o¤ between the precision of the infor-mation transmitted and the optimality of the policy implemented: precision is

6See Crawford and Sobel (1982).7Note that our approach remain valid if information campaigns are costly, as long as the

cost is independent of the message the authority decides to send. Since the di¤usion cost of“smoke is detrimental to one’s health” is the same as that of “smoke is very detrimental toone’s health”, we can, without generality loss, normalize this cost to zero, if no-message is nota choice.

8Actually the marginal cost of public funds can be either positive or negative, depending onthe relationship between preexisting taxes and the tax on the harmful good. Our model coversall the cases.

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higher with less e¢cient political programs, and conversely. We prove that theequilibrium is never e¢cient ex ante, and that there exists a unique fully-revealingequilibrium that is almost surely ine¢cient ex post.

Plan Section 2 presents the terms of the policy dilemma in the case of com-modities a¤ecting health and the environment. Section 3 de…nes the equilibrium.The main body of the paper develops the methodology. After a few results onthe structure of the government’s preferences (Section 4), the analysis is devel-oped in three steps. First we show that an equilibrium can be summed up by its“skeleton”, i.e. a relatively small set of policies satisfying incentive compatibilityfor the sender (Section 5): Second, we show under what circumstances a given“skeleton” can be implemented in an equilibrium. This is crucial to getting in-sights into the structure of partially revealing equilibria (Section 6). Finally wecharacterize the game’s unique fully-revealing equilibrium (Section 7). Some im-plications concerning tax policy for fuels, SO2 emissions and drugs are discussedin the conclusion.

2 The Model

2.1 The Consumers

Consumers live two periods and the value of their period-2 consumption x2 isnegatively a¤ected by period-1 consumption x1. Preferences can be written as:

U [x1] + x2 ¡ µx1 ¡ ´x1(1)

where U is the logarithmic utility function.9 The consequences of x1 on period-2utility pass through two distinct channels:

² The term ¡µx1 measures side e¤ects due to the consumer’s own consump-tion in period 1. The intensity µ is not precisely known to consumers. Thecumulative distribution function F (µ) and its density f(µ); both supportedin [µ; µ]; represent consumers’ priors on µ: In general, f is continuous andnon-negative on the support.

² The term ¡´x1 indicates the negative externality that depends on x1; i.e.average period-1 consumption in the economy. The intensity ´ is supposedto be known to all the agents.10 The consumer does not internalize the

9Most of the propositions in Section 4 (characterization of the equilibria) do not rely onthis restriction on utility, as can be seen in the proofs, but it does improve the legibility of theexplicit calculations of the …rst- and second-best.

10An alternative model could put uncertainty on ´: In general, though, this uncertainty alonewould not exhibit the sort of con‡ict we are pointing at, since the consumer’s behavior is not

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social consequences of x1. This is because there are a large number ofatomistic consumers in the economy: each knows that he would a¤ect theexternality only marginally.

Let t be the tax rate set by the government. The representative consumer,not internalizing the externality ´; solves:

(maxx1;x2

E [U [x1] + x2 ¡ µx1]s.t. : (p1 + t)x1 + p2x2 = W

(2)

where the expected value of utility is conditional on the consumer’s information;p1 and p2 are the prices for, respectively, period-1 and -2 consumption, and W isthe consumer’s endowment.

To further simplify the program we normalize p2 to 1 and p1 to 0 without lossof generality since the support of µ can be translated to account for the price,which is exogenous: Then we substitute the budget constraint into the objectivefunction and drop the subscripts to write the …rst period consumption as x: Thesimpli…ed consumer’s program is:11

maxxE [U [x]¡ (µ + t)x](3)

As a consequence, consumption choice x¤ depends on the consumer’s informationand on the tax rate t:

x¤[t; Eµ] solves U 0[x] = Eµ + t(4)

that isx¤[t; Eµ] =

1

Eµ + t(5)

2.2 Social Welfare and the Marginal Cost of Public Funds

The social welfare function that is maximized by the government corresponds tothe consumers’ utility once the externality and the exact value of side e¤ects aretaken into account. The revenue from income-, capital-tax, or other levies is ex-ogenous and taxation is distortionary. Within this public …nance perspective, wecalculate how x should be taxed. In addition to the fact that taxes are imperfect,the government is hampered by its inability to commit to a policy that informstruthfully on the value of µ. In other words, the government is benevolent in thatit evaluates consumption in consumers’ best interest, but opportunistic because

a¤ected by the intensity of the externality. In our speci…cation, consumption does not evendepend on ´:

11Notice that the linear part in the preferences also represent the utility from goods otherthan x:

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it does not value truthful information per se, and would deceive consumers pro-vided this induces “better” behavior (and reduces the distortions caused by thetax): in short it practices a variety of paternalism.

All consumers being identical, in equilibrium ¹x = x, and the government’sobjective function can be represented as:

U [x]¡ (µ + ´)x+ S ¡ (1 + ¸)R(6)

where S is the consumer’s surplus from public expenditures R: The governmentraises R with general taxation at the welfare cost (1 + ¸)R; with ¸ > ¡1: Mostpartial equilibrium models call parameter ¸ the ”shadow cost” of public funds;it represents the distortion due to the raising of …scal revenue.12

In our model, R and S remain constant, while a new tax on the good x isadded to existing taxes. As modern public …nance theory has shown, no generalconclusion can be drawn about the sign of the shadow cost of taxation when arevenue-neutral substitution between di¤erent taxes is implemented. The sign of¸ is not restricted a priori and depends on the structure of preexisting taxes (inparticular their e¢ciency) and on how they interact with the new tax.13

When the government introduces a tax t on good x and revenue tx is devotedto reducing preexisting taxes, (6) becomes, after simpli…cation:

U [x]¡ (µ + ´ ¡ ¸t)x+ S ¡ (1 + ¸)R(7)

Comparing (3) and (7), we see that the government’s program di¤ers from theconsumer’s in three ways: superior information on µ; internalization of ´; andthe presence of ¸ in the government’s objective function. As for the externality,the consumer does not internalize the e¤ect of his contribution tx on the totaldistortion caused by taxation.

The case ¸ > 0 (preexisting taxes in‡ict a welfare cost larger than R) relatesto the recently debated “double dividend” e¤ect. According to this literature,a revenue-neutral substitution of environmental taxes for ordinary income taxesmight o¤er a double dividend: not only does it (1) improve the environment butit also (2) reduces the costs of the tax system through cuts in distortionary taxes(see Goulder 1995). To intuit this result, assume that x and the other taxed goods(labor included) are gross substitutes. In that case, typically, the tax t reducesthe consumption of x and increases the consumption of the other taxed goods.Thus total …scal revenue increases and taxes on the other goods can be reduced,

12See, for example, the shadow cost of public funds used in the theory of regulation (La¤ontand Tirole 1993). In general equilibrium models of taxation (e.g. Ramsey), ¸ would be the(endogenous) Lagrange multiplier associated with the government’s budget constraint. Undersome regularity conditions, the Lagrange multiplier is equivalent to what the theory of cost-bene…t analysis calls the shadow cost of a marginal change in a public project. See Drèze andStern (1987).

13For a concise discussion, see Ballard and Fullerton (1992) and Goulder (1995).

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which attenuates distortions.14 ;15 Notice that we could also reason in terms ofrelative e¢ciency: when ¸ > 0 a tax on good x is relatively less distortionarythan preexisting taxes; when ¸ < 0; it is more distortionary.

Dropping constant terms that are not relevant to policy decisions, we get areduced form of the government’s objective function:

SW [x; t; µ] ´ U [x]¡ (µ + ´ ¡ ¸t)x(8)

2.3 Constrained E¢cient Allocations

The …rst-best allocation is de…ned as the allocation that maximizes consumers’utility when µ is known, ´ internalized, and the economy is not distorted (¸ = 0):This gives xFB(µ) =

1µ+´

(see equation (3) with ´ instead of t). This allocationcan be implemented even without full control over x with the standard Pigoviantax t = ´:

The second-best allocation is de…ned as the best the government can attain ifconsumers are perfectly informed on µ when (1) it is constrained to a linear taxon x; and (2) the marginal cost of public funds is not zero. This can be writtenas follows: (

maxtU [x]¡ (´ + µ ¡ ¸t)x

s.t. : x = 1µ+t

(9)

where the constraint on x corresponds to consumers’ reaction function (5) whenµ is known. Hence the second-best consumption and tax rate are:

½xSB(µ) =

1´+(1+¸)µ

tSB(µ) = ´ + ¸µ(10)

Because of the “double dividend”; the second-best tax is higher than the …rst-bestwhen ¸ > 0. The opposite holds for ¸ < 0: For a given ¸; the tax rate is strictlyincreasing (decreasing) with respect to µ when ¸ > 0 (¸ < 0): In any case, it isimportant to note at this step that the tax rate is potentially informative on thevalue of side e¤ects.

Straightforward calculations lead to a sort of Ramsey-Boiteux pricing rule:16

tSB(µ) ¡ ´1+¸

tSB(µ)=

¸

1 + ¸

1

"(11)

14Complementarity between x and the other taxed goods allows the same reasoning to holdwhen ¸ < 0.

15The same reasoning in terms of substitutability and complementarity between x; the othergoods, and the public project (…nanced by R) applies. In other words, the cost of public fundsalso depends on the interaction between the public expenditures and the taxed activities.

16A similar expression can be found in Sandmo (1975). See also Bovenberg and van der Ploeg(1994).

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where " ´ ¡@x¤@t

tx¤ =

tµ+t

is the tax elasticity of demand. As tax elasticity isdecreasing in µ for all t; (11) shows that for positive ¸; the stronger the sidee¤ects, the higher the tax. For negative ¸, the opposite holds:

From (8), and for a given tax t; the marginal net external e¤ect of consumptionx is ´ ¡ ¸t; where ´ is for external e¤ects sensu stricto, and ¡¸t for the e¤ectsof the tax on public …nance that the consumers do not internalize. Assumethat ¸ > 0: First, the social welfare function (8) shows that, others things (inparticular x) equal, the government would prefer to impose a “high” (in factin…nite) tax, in order to engender large positive externalities. Second, givensuch positive externalities, the government must decide how to tax x in orderto make consumers internalize them; according to the Pigou rule jt¡ (´ ¡ ¸t)jshould be minimized, and this is attained at t = ¹t ´ ´

1+¸. Unfortunately, these

two arguments draw in opposite directions and the two goals rely on the sameinstrument, t. This explains why the optimal trade-o¤ is tSB(µ); a value between¹t and +1.17

3 The In‡uence GameThe timing of the model is as follows: …rst the government observes µ; then itchooses its policy, and …nally the consumer, observing the policy, updates hisbeliefs on µ and chooses his consumption level.

A policy P = (t;m) 2 IR £M is composed of the tax rate t and a (cheaptalk) “message” m selected from a certain large set, M . Through the choice ofa policy P , the government wants to induce the consumer to approach e¢cientconsumption. The tax has the two-fold role of incentive and information; cheaptalk can only transmit information. We can think of m as composed of a “sen-tence”. We assume that M is broad enough to say what needs to be said; itmight be composed, say, of all reasonably short utterances (see, e.g., Farrell andRabin 1996 on what cheap talk is and is not). It is useful, at this point, to makea distinction between the message the government sends and the consumers’ in-terpretation of it at equilibrium. What really matters is not the message itselfbut the way the consumer understands the policy. To be clearer, whatever thephrasing of the communication, we concentrate on the meaning (the revised Eµ)that the consumer assigns to every policy.18

After observing the policy, the consumer updates his priors which are thendenoted by ¹(P ) (with ¹(P ) 2 ¢([µ; µ]); the set of probability distributions over

17When ¸ < 0; with …xed x, the government would like large subsidies (t ! ¡1); tSB (µ) isbetween ¡1 and t:

18As an example, let m1 and m2 denote two messages sent in a fully-revealing equilibrium.Assume that m1 corresponds to the word “dog” and m2 corresponds to the word “cat”. Thisis an equilibrium as long as the receiver understands this language and assigns to the message“dog” the meaning, say “µ = µ1”; and to the message “cat” the meaning, say “µ = µ2”; whereµ1 and µ2 2 [µ; µ]:

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[µ; µ]): We denote E(µjP ) by bµ(P ):

De…nition 1 A perfect Bayesian equilibrium (PBE) of the game is a pure strat-egy P mapping [µ; µ] into IR+ £M and a belief ¹ mapping IR£M into ¢([µ; µ])such that:

1. Policies are optimal given beliefs: for each µ 2 [µ; µ]; P(µ) solves

maxPSW [x¤[t;bµ(P )]; t; µ]:(12)

2. Beliefs are rational given equilibrium policy: for each P , x¤[t;bµ(P )] solves

maxx

Z µ

µ

[U [x]¡ (µ + t)x]¹(µjP )dµ;(13)

where ¹(µjP ) ´ IIfP(µ)=Pg¢f (µ)R µµ IIfP(µ)=Pg¢f(s)ds

; II being the indicator function.

The concern that revised beliefs may not always be well-de…ned is dealt within Proposition 3, below.

4 Government Policy Preferences

Bad news for communication gurus: the consumer’s rationality prevents the gov-ernment from transforming lead into gold by clever communication strategies. Inother words, we preclude any perverse mechanism whereby propaganda can makeless desirable states of the world (larger side e¤ects) preferable.

Proposition 1 In any equilibrium, the larger the side e¤ects, the lower the socialwelfare:

Proof. Let µ1 and µ2 be two possible states of the world, P1 = (t1;m1) andP2 = (t2;m2) two equilibrium policies, and x1 and x2 the consumption levelsinduced. If µ1 < µ2; then U [x2] ¡ (´ + µ1 ¡ ¸t2)x2 ¸ U [x2] ¡ (´ + µ2 ¡ ¸t2)x2:On the other hand, the incentive constraint of the type-µ1 social planner reads:U [x1] ¡ (´ + µ1 ¡ ¸t1)x1 ¸ U [x2] ¡ (´ + µ1 ¡ ¸t2)x2: By transitivity, we get:U [x1] ¡ (´ + µ1 ¡ ¸t1)x1 ¸ U [x2] ¡ (´ + µ2 ¡ ¸t2)x2: Thus the social planner’spay-o¤ decreases with respect to the side e¤ects.¥

Now let us analyze the government’s incentive to manipulate information:that is, the reasons why consumers are likely to be suspicious of the government’sactions and claims, and how suspicious.

Remark 1 In equilibrium, any policy P can be analyzed without loss of insightas a pair (t;bµ), where t is the tax rate, and bµ the belief associated to the policy.

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We de…ne SW [t;bµ; µ] ´ SW [x¤[t;bµ]; t; µ] as the value of a policy characterizedby the tax-belief pair (t;bµ) for a government of type µ: Reasoning directly on tax-belief pairs allows a simpler analysis of incentive constraints, independently of thecheap talk message chosen. Indeed, incentive compatibility for P (µ) = (t;bµ) andP (µ0) = (t0;bµ0) can clearly be checked by comparing SW [t;bµ; µ] with SW [t0;bµ0; µ];and SW [t;bµ; µ0] with SW [t0;bµ0; µ0]:

The consumer solves U 0[x] = bµ(P ) + t: Therefore from (8), we can see thatthe consumer’s choice is equal to the socially optimal consumption when:

bµ(P ) + (1 + ¸) t = µ + ´(14)

Suppose that the consumer is naive and believes whatever the governmentannounces. The government would set the tax rate and induce beliefs so that(14) is veri…ed. Note that the right-hand side of (14) is a constant. When ¸ > 0(t increases social welfare), the government prefers to set high taxes and to inducelow beliefs. In other words, it prefers to make the consumer optimistic about sidee¤ects, and relies mostly on taxation. The opposite is true when ¸ < 0 (t yieldsdeadweight losses): the government prefers to make the consumer pessimisticabout side e¤ects and to drive taxation to its lowest level.

Vis-à-vis a rational consumer, such a policy is obviously never consistent;nevertheless, it provides useful indications on the incentives that the governmentperceives. For instance, when only cheap talk is available, equation (14) becomesbµ(P ) = µ+ ´: In this case, the government always has incentives to overstate thevalue of µ so as to make the consumer internalize the externality. The setting thenresembles Crawford and Sobel (1982), which as we saw earlier, explains why thehealth authorities are better o¤ when consumers have an exaggerated perceptionof the side e¤ects of antibiotics.

Now we become more formal. Policies are restricted to induce …nite consump-tion. Thus feasible policies are such that t + bµ > 0. The di¢culty here is thatindi¤erence curves are not monotonic: there is an optimal policy (unfortunatelyinconsistent with Bayesian consumers, as we shall show), and utility decreases asthe tax and the belief get farther from the optimum. Nevertheless, the follow-ing proposition gives us some useful properties to proceed with the analysis ofincentive compatibility.

Proposition 2 1. For all µ; the upper contours of SW with respect to t andbµ are convex.

2. For all µ; tangents to indi¤erence curves are horizontal along the straightline (1+¸)t+bµ = ´+µ; and vertical along the straight line t+(1¡¸)bµ = ´+µ:The overall optimum is the intersection of these lines (t = ´+µ

¸;bµ = ¡´+µ

¸);

the optimum with t = 0 is bµ = µ + ´.

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3. Let V(µ) be an indi¤erence curve for type µ passing through (t;bµ). V(µ)turns continuously clockwise if ¸ > 0 (anti-clockwise if ¸ < 0) locally at(bµ; t) as µ increases and indi¤erence curves related to two di¤erent typescross once at most.

Proof. 1. It su¢ces to verify that the utility is quasi-concave. To do this, wecheck that the successive principal minors of the bordered Hessian matrix havealternate signs (odd principal minors must be positive). The bordered Hessianmatrix is:

26664

0 ´+µ¡t¡(1¡¸)bµ(t+bµ)2

´+µ¡(1+¸)t¡bµ(t+bµ)2

´+µ¡t¡(1¡¸)bµ(t+bµ)2 ¡2´+2µ¡t¡(1¡2¸)bµ

(t+bµ)3 ¡2´+2µ¡(1+¸)t¡(1¡¸)bµ(t+bµ)3

´+µ¡(1+¸)t¡bµ(t+bµ)2 ¡2´+2µ¡(1+¸)t¡(1¡¸)bµ

(t+bµ)3 ¡2´+2µ¡(1+2¸)t¡bµ(t+bµ)3

37775(15)

The …rst principal minor is equal to zero, the second is negative, and we …nd¸2

(t+bµ)4 for the third, which gives result required.

2. and 3. The MRS between t and bµ is

dt

dbµ

¯̄¯̄SW=constant

= ¡´ + µ ¡ (1 + ¸)t¡ bµ´ + µ ¡ t¡ (1¡ ¸)bµ

(16)

Its derivative with respect to µ is

¡ ¸(t+ bµ)(´ + µ ¡ t¡ (1¡ ¸)bµ)2

(17)

which is negative (positive) for ¸ > 0 (¸ < 0) for t+bµ > 0:Tangents to indi¤erencecurves being vertical whenever ´+ µ¡ t¡ (1¡¸)bµ = 0, the claim is correct. Theoptimum is the singular point where both the numerator and the denominator of(16) are equal to zero.

Notice that if one sets aside domain restrictions, upper contours are closed,meaning that two indi¤erence curves related to two di¤erent types, if ever theycross, cross twice at least. Since we have proved here that in the relevant range(t+ bµ > 0) crossing occurs once at most, the standard argument based on singlecrossing can be retrieved.

Figure 1 shows the government’s indi¤erence curves for ´ = 1; µ = 0; ¹µ = 1;¸ = :7 and µ = 1.

Insert Figure 1 here.

We now can prove that the second-best policy is not implementable in aBayesian equilibrium. Suppose the consumer thinks that the government is play-ing the second-best strategy. The tax schedule tSB being invertible; if tSB(µ) is

13

imposed, the individual can infer µ unambiguously. Nevertheless, it is not pos-sible to implement this allocation in a Bayesian equilibrium. In fact, the …scalrevenue tSB(µ)xSB(µ) increases as µ decreases. When ¸ > 0 and µ is high, thegovernment may have an interest in reporting a lower µ; in other words in mak-ing the consumer optimistic. On the contrary, when ¸ < 0; the government mayhave interest in making the consumer pessimistic. Proposition 1 con…rms thatthe government faces strong incentives to provide biased information.

Corollary 1 The second-best allocation is never an equilibrium if ¸ 6= 0:

In fact, at (tSB(µ); µ) = (´ + ¸µ; µ); for all µ; the tangent of the indi¤erencecurve of the government is vertical (see point 2 in the Proposition): small changesin the tax have …rst-order e¤ects, whereas small changes in the beliefs have onlysecond-order e¤ects on the government’s objective. Consequently, if ¸ > 0; anypolicy close to the second-best (´ + ¸µ; µ) but with t < ´ + ¸µ is preferred; thisholds for a second-best policy associated with a slightly type. If ¸ < 0; second-best policies associated with slightly higher types are preferred. In any case,the second-best allocation is not incentive-compatible, which con…rms that thegovernment has a strong incentive to bias its information campaign.

Note in contrast that when ¸ = 0; the …rst-best allocation is implementablein a PBE. Indeed, given that the government has no incentive to lie (see (14)),the tax is speci…cally used to internalize the externality (t = ´), but since thetax rate is uninformative on µ, cheap talk must be used to eliminate asymmetricinformation. With a slight abuse, equilibrium policies can be written as P =(t = ´;bµ = µ); where information is fully transmitted. This is, of course, a veryparticular case.

5 Skeletons

Description of all the equilibria given the prior type distribution is di¢cult. Sorather than look for equilibria in the traditional way for signalling games, weintroduce a di¤erent technique. That is we solve the inverse problem: we …ndthe set of types and the distributions of types that are consistent with a certainequilibrium allocation. This new approach has some relationship with mathe-matical tools mostly used in imagery (namely Voronoi diagrams, and their dual,Delaunay triangulation) from which we borrow our terminology (the skeleton).19

The analogy is the following: given a partition of the types, types in each subsetapplying the same (unknown) policy, and two di¤erent subsets applying di¤erent

19The idea is basically the following: the Voronoi diagram of a point set P is a subdivisionof the plane with the property that the Voronoi cell of point p contains all locations that arecloser to p than to every other point of P. The points of P are also called Voronoi generators.Each edge of a Voronoi cell is the bisector of the connection of p to the corresponding neighbourcell. See http://www.voronoi.com/ for theory, algorithms, and examples of applications.

14

policies, one may want to inquire into the underlying policies. Conversely, givena certain set of policies, and given that the government responds to its incentives,one may be interested in the types that have to be associated with each policy.In all these problems, preferences can be seen as a measure of distance.

The following proposition generalizes the well-known result (Crawford andSobel 1982, henceforth CS) that all equilibria are “partition equilibria”. See alsoAusten-Smith and Banks (2000).

Proposition 3 Any equilibrium allocation can be implemented in an equilibriumin which there exists a partition of [µ; µ] into a set of intervals fIkgi2K (K isa minimal set of indices) and a set of policies fPkgk2K such that (i) the policychosen in Ik is Pk; and (ii) k 6= k0 implies Ik 6= Ik0 and Pk 6= Pk0: Moreover,the e¤ects of policy Pk are entirely characterized by the pair (tk;bµk), where bµk ´E(µjPk) = E(µjIk)); 8k:

Proof. In this proof, optimal is used in the weak sense. In any PBE, for allP that are equilibrium actions, the set of types for which P is optimal is aconvex subset of [µ; µ]: To see this, consider the sender’s incentive constraint in agiven equilibrium. Type µ will prefer policy P1 = (t1;m1) to any P2 = (t2;m2);implying, respectively, consumptions x1 and x2, if and only if:

U [x1]¡ (´ + µ ¡ ¸t1)x1 ¸ U [x2]¡ (´ + µ ¡ ¸t2)x2 ,(18)

µ(x2 ¡ x1) ¸ U [x2]¡ U [x1] + ´(x1 ¡ x2) + ¸t2x2 ¡ ¸t1x1

This equation de…nes either a half straight-line in the space of types (x1¡x2 6= 0)or the whole real line (x1 = x2). From this, it follows that if policy P is optimalfor two values of µ; then it is optimal for any type that lies between these twovalues.

Let us denote by (µ1; µ2); with µ1 6= µ2; an interval in which P1 is optimal.We now check that there is only one optimal policy in the interval. Suppose, forpurpose of argument, that this is not the case, e.g. 9µ 2 (µ1; µ2) for which bothP1 and P2(6= P1) are optimal. Equation (18) thus becomes

µ(x2 ¡ x1) = U [x2]¡ U [x1] + ´(x1 ¡ x2) + ¸t2x2 ¡ ¸t1x1(19)

Now either x1 6= x2, and, according to (18), P1 is strictly preferred to P2 onone side of µ; and P2 is strictly preferred to P1 on the other side, which is incontradiction with our assumption that P1 is optimal on (µ1; µ2); or else x1 = x2;which implies in turn that t1 = t2; and (given that consumptions are only afunction of the tax and the beliefs) that P1 and P2 imply the same beliefs. Inthis case, P1 and P2 are the same in terms of tax and beliefs. Though they maydi¤er in their cheap talk dimension, they can be seen as identical, and Remark 1shows why.

15

If an equilibrium allocation were not implementable by a strategy based on apartition into intervals, then the latter result would be false. Hence, our claim isproven.

One substantial implication of this result is that Condition 1 in the de…nitionof the equilibrium (Section 3) implies the non-evident property that beliefs inCondition 2 are well-de…ned (indeed, strategies inherit the measurability of thespace of types).

Proposition 3 suggests that only a “few” tax-belief pairs are interesting. Wecan go further and show that only a “few” incentive-compatibility constraintsneed to be checked to ensure that an allocation is an equilibrium.

De…nition 2 (Skeleton) Let fµkgk2K be a closed subset of [µ; µ] in which k 6=k0 implies µk 6= µk0 (K is a minimal set of indices); and let ftkgk2K be a setof real numbers. f(tk; µk)gk2K is said to be a skeleton if and only if 8k; k0 2K;SW [tk; µk; µk] ¸ SW [tk0; µk0; µk] (incentive compatibility).

One particularity of the skeleton is that any equilibrium to which it is con-nected is revealing for the types of the skeleton, and for these types only (this isrepresented by the incentive constraints in the de…nition of skeleton).

Proposition 4 below is the reciprocal of Proposition 3. We exploit the ideathat the skeleton represents the essential data that characterize an equilibrium.The type support can be divided into intervals in which the strategy is pooling,and we specify the restrictions on the “‡esh” (the distribution F ) that may beput on the “bones” (the skeleton) to obtain an equilibrium.

Proposition 4 Let F be the set of type distribution F such that the skeletonf(tk; µk)gk2K is an equilibrium set of policies. There exists a partition of [µ; µ]into a set of intervals fIkgk2K with µk 2 Ik such that: 8F 2 F ; 8k; t(¢) = tk overIk and E(µjIk) = µk:

Proof. By convention, we denote the lowest element of fµkgk2K as µ1; and thehighest as µ1:Given µk; we de…ne its successor in fµkgk2K as µk+1 ´ mink02Kfµk0 >µkg (this “+1” is just a convention, inspired by the fact that when K is …nite, itcan be reformulated as a set of successive integers). The type µk+1 is well de…nedsince a skeleton is closed.20 We reason on incentive compatibility.

If µk+1 6= µk; we denote by ¿ k a type which is indi¤erent between P (µk)and P (µk+1); i.e. SW [tk; µk; ¿ k] = SW [tk+1; µk+1; ¿k]: Given the single crossingproperty, and given the continuity of the government’s welfare function withrespect to the true type, ¿ k is unique and belongs to [µk; µk+1]: We de…ne Ik =

20Notice that we assume that fµkgk2K is closed only to simplify our reasoning. This assump-tion is in facti without loss of generality: if an accumulation point of fµkgk2K were missing (i.e.if fµkgk2K were not complete), we could add it to fµkgk2K , with a corresponding accumulationpoint in ftkgk2K : Due to the continuity of the incentive constraints, incentives are not reversed,and the skeleton is completed.

16

(¿ k¡1; ¿ k]: If the successor of µk is µk itself (this happens if µk is, on the right, anaccumulation point in fµkgk2K), then Ik = fµkg: The lower bound of the lowestinterval (i.e. containing µ1) is µ, and the upper bound of the upper interval(containing µ1) is µ: Given Proposition 3, t(¢) = tk over Ik for all k is incentive-compatible. Finally, to ensure that the equilibrium beliefs of the consumer areBayesian, it is necessary and su¢cient that F (¢) be such that E(µjIk) = µk:

Conditional expectations (with respect to the policy, or to the interval) areindependent of one another. The probability associated with the interval Ik notbeing constrained, F 2 F can be chosen to be as smooth as wanted.

Figure 2 summarizes the notation and the main properties of the skeleton.

Insert Figure 2 here.

Corollary 2 If two di¤erent intervals are associated with two di¤erent tax rates,then the tax rate is su¢ciently informative for the consumer, and the messagecan be ignored. If there exists k 6= k0 such that tk = tk0; then messages areindispensable to signal the right interval and ensure the right beliefs.

When the tax rate is the same for two or more intervals, cheap talk servesto transmit some information. In the terminology of Austen-Smith and Banks(2000), cost-free signalling is in‡uential if two di¤erent cheap talk messages as-sociated with the same tax rate have to be used to distinguish two di¤erentintervals.

Similarities with CS are obvious: Propositions 3 and 4 show that the govern-ment can use meaningful yet imprecise policies to communicate the side e¤ectsto consumers. Since the government has interest in lying on the value of µ tomake consumers internalize the externality, powerful communication campaignswould give the government the means of manipulating consumers’ beliefs. Asa consequence, the government is restricted in equilibrium to vague statementsthat only specify broad ranges within which µ may lie.

This trade-o¤ is classical for readers accustomed to cheap talk: the partitionfIkgk2K entails a loss in precision, but now, if the government wants to lie, ithas to pretend that the side e¤ects are in a di¤erent subinterval, which changesconsumers’ consumption by a discrete amount. Such “big lies” are less attractivethan telling the truth.

A less evident conclusion is that there are also considerable di¤erences fromCS. The skeleton approach enables us to show that partitions need not be …nite,meaning that the precision of the message may be arbitrarily high locally. In thissense a new trade-o¤ arises: as precision increases, tax policies are more severelyconstrained by incentive compatibility, and distortions away from the second-bestbecome large.

17

6 Partially Revealing Equilibria

We can start to build an equilibrium by choosing a skeleton and …lling the dis-tribution while preserving conditional expectations. Proceeding in this way, wereadily provide examples in which the tax rate is not monotonic, where it is reveal-ing on certain subsets of [µ; µ] with bundles elsewhere, etc. Hence a multiplicityof partially revealing and pooling equilibria are conceivable.

Proposition 4 does not claim that some distribution F always exists. Indeed,even o¤-equilibrium beliefs are constrained to be in [µ; µ], and we may be shortof su¢ciently dissuasive o¤-equilibrium beliefs to support a skeleton. We arenevertheless able to furnish a simple way of extending a skeleton to make F nonempty, in other words, to implement the skeleton in a PBE.

Proposition 5 Any skeleton is either directly implementable or can be madeimplementable by adding one policy (one belief and its associated tax).

Proof. Let us take an un-implementable skeleton. If ¸ > 0; the simplest way tocomplete it is to add a su¢ciently low type, say µmin < µ1; coupled with tSB(µmin).This may mean enlarging [µ; µ] by replacing µ with µmin: If we associate beliefµmin with any tax outside ftkgk2K ; we still have a skeleton. To check incentivecompatibility, note that if belief µmin is su¢ciently small compared to µ0; it isnecessarily too small compared to any type of government drawn in fµkgk2K .Moreover t = tSB(µmin) is better than any other value of the tax for a governmentof type µmin: The new skeleton is now implementable. If ¸ < 0; the same reasoningholds for a large µmax > µ1 (coupled with tSB(µmax)).

This suggests that, as long as priors are de…ned over a su¢ciently large set,and even if extreme types are extremely unlikely, one may exploit the presenceof “scarecrow” types to build equilibria.

Our model di¤ers signi…cantly from pure cheap talk models. Indeed, with…nite skeletons, cheap talk doesn’t really need to be in‡uential (i.e. useful), sinceeither all taxes are di¤erent or some are identical and we can use the continuityof the incentive constraints to modify the skeleton slightly and make all tax ratesdi¤erent, in which case cheap talk is useless. In other words, suppose that (tk;mk)and (tk0;mk0) are two equilibrium policies with tk = tk0 = t and mk 6= mk0 (cheaptalk is in‡uential). If we change one of the two taxes, the partition in the skeletonhas to be modi…ed, but changes remain small because there are only a …nitenumber of bones (hence a …nite number of continuous incentive constraints) andthe welfare cost of changing is arbitrarily low. The extension of this intuition to alarge set of signals is not developed here. See Manelli (1996) for another approachto the same sort of result (i.e. cheap talk closes but does not substantially extendthe set of equilibrium allocations). The previous reasoning shows that the roleof cheap talk as stated in Proposition 4 is sharply diminished, since in a verystrong sense cheap talk is almost useless when a costly message (here the tax) isavailable.

18

7 Fully Revealing Equilibria

By de…nition, fully informative equilibria have exhaustive skeletons in which alltypes are represented. Moreover, Proposition 4 implies that the correspondingallocation is a universal skeleton, that is an equilibrium for any distribution Fin [µ; µ]: The following proposition establishes that for any given [µ; µ] there isa unique fully revealing equilibrium (or a unique universal skeleton), which wecharacterize in detail.

Proposition 6 There exists a unique fully revealing equilibrium. The tax rate

is the unique solution to the ordinary di¤erential equation t01+¸

= ¡ t¡ ´1+¸

t¡tSB (µ)with

the boundary condition t(µ) = tSB(µ) if ¸ > 0; and t(µ) = tSB(µ) if ¸ < 0: Inparticular:

1. Cheap talk is ine¤ective, and the strategy t(¢) is strictly increasing and dif-ferentiable.

2. Consumption decreases with respect to µ:

3. If ¸ > 0; the tax rate exhibits no distortion at µ: For other values, the taxrate is lower than the second-best tax rate and higher than ´

1+¸:

4. If ¸ < 0; the tax rate exhibits no distortion at µ: For other values the taxrate is higher than the second-best tax rate and lower than ´

1+¸

Proof. See the Appendix.On the role of cheap talk, it is clear from point 1 that in the fully informative

equilibrium all the information is transmitted through the tax rate. When ¸ > 0(¸ < 0); the fully informative equilibrium allocation, compared to the second-bestone, is characterized by taxes that are too low (or too high). Moreover, when¸ > 0 ( ¸ > 0) taxes are decreasing (increasing) with respect to the type.

The di¤erential equation assigns essential roles to the second-best tax and tot = ´

1+¸: The government (though under incentive constraints) tries to maximize

welfare, hence to approach tSB(µ) as nearly as possible. On the one hand, thecloser the tax is to tSB(µ), the higher social welfare, but the stronger the incentivesfor lying and the steeper the slope of the revealing tax schedule. On the otherhand, for tax rates approaching the suboptimal t; incentives to manipulate beliefsvanish, and the revealing tax schedule ‡attens.21

Here lies the origin of the distortion: credibility is gained by moving away fromthe optimal schedule. Note that if the government could commit ex ante to tax twhatever the state of nature µ; then telling the truth by means of cheap talk wouldbe sequentially optimal since no credibility problem would arise. Unfortunately,this easy credibility would come at the price of a severe lack of e¢ciency!

21On the properties of t; see Subsection 2.3.

19

Figure 3 shows the fully revealing tax rate for ´ = 1; µ = 0; ¹µ = 1 and ¸ = :3:Notice the indi¤erence curves passing through the equilibrium value for µ = :8and µ = 1. Figure 4 corresponds to ¸ = ¡:3 (other parameters are equal to thosein Figure 3). This illustrates the non-negligible size of the distortion.

Insert Figure 3 here.Insert Figure 4 here.

The limited role of cheap talk can also be viewed another way. As ¸ ! 0;one can …nd a sequence of fully-revealing equilibrium allocations converging onthe …rst-best where t = ´ for all µ: However, if we keep restricting the signal tobe supported only by the tax, the …rst-best is not an equilibrium since no preciseinformation on µ can be transmitted: at the limit, cheap talk is indispensable,but very close approximations in which it is not used are available.

In CS, the most informative equilibrium Pareto-dominates, the others exante.22 Austen-Smith and Banks (2000) …nd that this is not true when cheaptalk and burning money to signal the type are used together. With our skele-ton approach, it is relatively plain that the unique-fully informative equilibriumallocation need not be e¢cient with our approach in terms of skeletons. To seethis, take an equilibrium and take its skeleton. The substance of Proposition 4 isthat any economy that satis…es the restrictions on the conditional expected typein the intervals associated with the skeleton can implement it in equilibrium. Ifthe mass of an interval where the distortion is substantial is su¢ciently large,then the equilibrium is necessarily ine¢cient ex ante. More generally, given twoskeletons, one more informative than the other (a …ner partition in intervals),the less informative one can be made more e¢cient by choosing the distributionappropriately.

8 Conclusion

We have examined the con‡ict between providing incentives and transmittinginformation that arises when an informed and benevolent government combineslinear taxes and information campaigns. Our model suggests that the governmentmay have trouble gaining credibility for its actions and messages, even thoughits objective is aligned with the consumers’. The problem is that the governmentcannot commit to reveal information truthfully. Its instruments being imperfect,it has strong incentives to enhance their impact with providing biased informa-tion.

Depending on the kind of distortion that prevails in the …scal system (i.e.whether the tax generates a “double dividend” or not), the government would like

22See Theorems 3 and 5 in CS, which say that both the sender and the receiver strictly preferequilibrium partition with more steps.

20

to make consumers either pessimistic or optimistic about the e¤ect of consump-tion on individuals. In the likely case of positive marginal cost of public funds,if consumers were more optimistic, the government could set higher taxes with-out curbing consumption too severely, and the distortions created by preexistingtaxes could be easily alleviated. Fuel taxes are an example: the government maynot wish to stress the dangers of motor vehicles in order to preserve this readysource of revenue. The paradoxical consequence is that, at the fully revealingequilibrium, there is a bias towards excessively low taxes.

Our example of negative costs of public funds is quite informal. In France, SO2

emissions are subjected to a “para…scal” tax: an independent agency is in chargeof collecting the tax and also redistributes the proceeds in the form of subsidiesfor abatement e¤orts. Thus even if the Treasury has positive marginal costs ofpublic funds, the agency may have negative marginal costs. It may be temptingto exaggerate local e¤ects (represented by our µ in the agents’—here the …rms’—programs) to economize resources wasted in the costly collection/redistributionprocess. As a result, at the fully revealing equilibrium, paradoxically the agencyshould be biased towards exaggeratedly high taxes.23

Another policy implication of the model is that information campaigns à laCrawford and Sobel are practically super‡uous when the social planner can alsorely on costly signals. Quite simply, a costly tax is taken more seriously than merepropaganda, and information campaigns characterized by short phrases whosecontent is too vague to be veri…able (“smoking is harmful to health”) are often ofvery limited e¢cacy. This result is in line with the empirical evidence of Bardsleyand Olekalns (1999) on the impact of health warnings on cigarette packs. Tobe sure, we must distinguish here between information campaigns and “hardinformation”. The former take the form of “free” advertising while the latterimplies that the government collects and presents detailed scienti…c evidence insupport of its views and that it employs other relays (academics, teachers, socialworkers, newspapers, etc.), in hopes that credibility will cease to be an issue.This process is slower, but presumably more e¤ective.

References

[1] Austen-Smith, David (1990): “Information Transmission in Debate,” Amer-ican Journal of Political Science, 124–152.

[2] Austen-Smith, David and Je¤rey S. Banks (2000): “Cheap Talk and BurnedMoney,” Journal of Economic Theory, 91, 1, 1–16.

23As far as health policy is concerned, one can imagine that health autorities are driven todeliver cautious messages on the side e¤ects of antibiotics. In equilibrium, however, they mustseverely limit reimbursements (which is similar to imposing high taxex) to signal toxic drugs.

21

[3] Ballard, Charles L. and Don Fullerton (1992): “Distortionary Taxes and theProvision of Public Goods,” Journal of Economic Perspectives, 6, 117–131.

[4] Bardsley, P. and N. Olekalns (1999): Economic Record, 75, 230, September,225–40.

[5] Bovenberg, A. Lans and Frederick van der Ploeg (1994): “EnvironmentalPolicy, Public Goods and the Marginal Cost of Public Funds,” EconomicJournal, 104, 444–554.

[6] Brown, Gardner and David F. Layton (1996): “Resistance Economics: So-cial Cost and The Evolution of Antibiotic Resistance,” Environment andDevelopment Economics, 1, 3, 349–355.

[7] Crawford, Vincent P. and Joel Sobel (1982): “Strategic Information Trans-mission”, Econometrica, 50, 6, 1431–1451.

[8] Drèze, Jean and Nicholas Stern (1987): “The Theory of Cost-Bene…t Analy-sis”, in Handbook of Public Economics, II, edited by A.J. Auerbach and M.Feldstein, Elsevier Science Publishers.

[9] Farrell, Joseph and Matthew Rabin (1996): “Cheap Talk,” Journal of Eco-nomic Perspectives, 10, 3, 103–118.

[10] Goulder, Lawrence H. (1995): “Environmental Taxation and the DoubleDividend: A Reader’s Guide,” International Tax and Public Finance, 2,157–83.

[11] Grossman, Gene M. and Elhanan Helpman (2001): Special Interest Politics,forthcoming.

[12] La¤ont, Jean-Jacques and Jean Tirole (1993): A Theory of Incentives inProcurement and Regulation, MIT Press, Cambridge and London.

[13] Levy, Stuart B. (1992): “The Antibiotics Paradox”, Plenum Press, NewYork.

[14] Manelli, Alejandro M. (1996): “Cheap Talk and Sequential Equilibria inSignaling Games,” Econometrica, 64, 917–942.

[15] Maskin, Eric and Jean Tirole (1992): “The Principal-Agent Relationshipwith an Informed Principal, II : Common Values,” Econometrica, 60 , 1–42.

[16] Milgrom, Paul and John Roberts (1986): “Prices and Advertising Signals ofProduct Quality,” Journal of Political Economy, 94, 796–822.

[17] Sandmo, Agnar (1975): “Optimal Taxation in the Presence of Externalities,”Swedish-Journal-of-Economics, 77, 1, 86–98.

22

[18] Okeke, Iruka N., Laminkanra, Adebayo, Edelman, Robert (1999): “Socioe-conomic and Behavioral Factors Leading to Acquired Bacterial Resistanceto Antibiotics in Developing Countries,” Emerging Infectious Disease, 5, 1,18–27.

[19] Viscusi, W. Kip (1990): “Do Smokers Underestimate Risks?” Journal ofPolitical Economy, 98, 6, 1253-1269.

A Appendix

A.1 Proof of Proposition 6

We establish the result in two steps. The …rst analyzes di¤erentiable fully-revealing equilibria; uniqueness in this category is proved. The second step showsthat any fully-revealing equilibrium is essentially identical to the di¤erentiableone.

A.1.1 Di¤erentiable Equilibrium

The analysis follows this plan: reasoning on local incentive compatibility, we …ndthe ordinary di¤erential equation satis…ed by any fully-revealing equilibrium taxpolicy and we eliminate solutions with tax rates that do not fall between ´

1+¸

and the second-best schedule (whichever is the higher); we check global incentivecompatibility along the equilibrium policy; we search for o¤-equilibrium beliefs(i.e. associated with o¤-equilibrium tax rates) that discourage deviations. Thisgives a unique equilibrium.

Local Incentive Compatibility The government prefers t(µ) (and the impliedx(µ)) to t(µ + dµ) and to t(µ ¡ dµ); taking limits we get

x0U 0 + ¸t0x¡ (´ + µ ¡ ¸t)x0 = 0:(20)

Given that the consumer’s …rst-order condition is

U 0 = µ + t;(21)

we can eliminate U 0 to get (after simpli…cation)

tx0 = ¡¸t0x+ (´ ¡ ¸t)x0:(22)

t and x being separable, (22) is easily integrated to give

8µ; µ0 :´1+¸

¡ t(µ)´1+¸

¡ t(µ0)=

µx(µ)

x(µ0)

¶ 1+¸¸

(23)

23

where µ0 and t0 = t(µ0) are initial conditions. Equations (23) and (5) implicitlybut entirely determine the solutions t(µ) and x(µ): In particular, x solves

µ(µ +

´

1 + ¸)x¡ 1

¶x1¸ =

µ(µ0 +

´

1 + ¸)x0 ¡ 1

¶x1¸0 = Constant(24)

By di¤erentiation, we get

x0 =¸x2

1¡ (´ + (1 + ¸)µ)x(25)

The second-order condition is:

0 ¸ x02U 00 + x00U 0 + ¸t00x+ 2¸t0x0 ¡ (´ + µ ¡ ¸t)x00;(26)

while the derivative of the …rst-order condition is:

0 = x02U 00 + x00U 0 + ¸t00x+ 2¸t0x0 ¡ x0 ¡ (´ + µ ¡ ¸t)x00;(27)

Simplifying (26) with (27) we get:

x0 · 0(28)

Applying (28) to (25) and using (5), we can see that, when ¸ > 0; x0 · 0 if andonly if t < ´ + ¸µ and when ¸ < 0; x0 · 0 if and only if t > ´ + ¸µ:

Starting from (25) and (22), straightforward calculations prove that the dif-ferential equation satis…ed by t is

t0 =´ ¡ (1 + ¸)tt¡ ´ ¡ ¸µ :(29)

Global Incentive Compatibility The …rst- and second-order conditions ex-clude in…nitesimal deviations. We check that discrete deviations are also pre-cluded.

Let µ be the true value of the side-e¤ects parameter. Using (5) we calculatethe derivative of the government’s utility with respect to bµ assuming that thegovernment o¤ers t(bµ), thereby inducing x(bµ) :

x0(bµ)U 0[x(bµ)] + (¸t0(bµ)x(bµ)¡ (´ + µ ¡ ¸t(bµ))x0(bµ))(30)

=x0(bµ)x(bµ)

+ ¸t0(bµ)x(bµ)¡ (´ + µ ¡ ¸t(bµ))x0(bµ)

From (22), we …nd that the following expression has the same sign as (30)

x(bµ)(t(bµ) + µ)¡ 1(31)

Using (5), it follows that (30) is positive for bµ < µ and negative for bµ > µ: Thismeans that incentive compatibility is satis…ed everywhere for equilibrium actions.

24

Uniqueness of the Di¤erentiable Equilibrium We give the full reasoningfor ¸ > 0: A symmetrical argument proves the proposition for ¸ < 0: We shownow that the boundary condition t(µ) = ´ + ¸µ is necessary.

Reasoning by contradiction, we show that there exist beliefs compatible withthe equilibrium for o¤-equilibrium actions if and only if the condition is satis…ed.Let t(¢) be a solution to (29) such that t(µ) < ´ + ¸µ:24 Given (29), either t(µ)is systematically below ´

1+¸or t(µ) is strictly increasing. In any case maxµ t(µ) <

´+¸µ; we choose an arbitrary (o¤-equilibrium) t in the interval (maxµ t(µ); ´+¸µ)and we denote by bµ the associated belief. Now we prove that there always existsa type µ such that the government prefers policy (t,bµ) to policy (t(µ); µ):

From Proposition 2, we know that for each µ; the absolute best policy is ´+µ¸

for the tax rate and ¡´+µ¸

for the belief; moreover, the second-best (´ + ¸µ; µ)is preferred to (t(µ); µ): The convexity of the upper contours of the government’sobjective function implies that any policy in the triangle 4(µ) = ((t(µ); µ); (´ +¸µ; µ); (´+µ

¸;¡´+µ

¸)); except (t(µ); µ), is strictly better than (t(µ); µ) when µ is

the type. It su¢ces now to check that (t,bµ) is necessarily in 4(µ) for a cer-tain µ 2 [µ; µ]: Indeed, [µ2[µ;µ]4(µ) contains (a) the triangle ((´ + ¸µ; µ); (´ +

¸µ; µ); (´+µ¸;¡´+µ

¸)); and (b) the policies between (t(µ); µ) and (tSB(µ); µ) for µ 2

[µ; µ]: Provided ´+µ¸

is larger than ´ + ¸µ; then (t,bµ) is either in (a) or (b) in thelatter union, hence the existence of a µ for which the deviation is desirable. Giventhat ´+µ

¸> ´ + ¸µ; the proof is complete. The consequence is that t(µ) = ´ + ¸µ

(no distortion at the top).Now we prove that associating belief µ to any tax rate above t(µ) does not

induce deviations. The value to the government of type µ of imposing t > t(µ);thereby inducing belief µ; is: ¡ log(µ + t) ¡ ´+µ¡¸t

µ+t: The root of the derivative

with respect to t is ´+ µ¡ (1¡¸)µ which is lower than ´+ ¸µ = t(µ): The valuebeing decreasing with respect to t over [t(µ);+1[, t(µ) is a better move than anyt > t(µ): Given that equilibrium actions are incentive-compatible, neither t(µ)nor t is desirable, compared to t(µ). By the same reasoning, we can check that,if for t < t(µ); beliefs are µ; then t is not attractive: for a government of type µ;the value of imposing t < t(µ) is smaller than that of imposing (t(µ); µ):

We conclude that the unique revealing allocation found is an equilibrium.

A.1.2 Uniqueness in General

Let us take a fully-revealing equilibrium. Given that the government’s prefer-ences, for constant beliefs, are single-peaked with respect to t (a direct conse-quence of the convexity in Proposition 2), and given the value of its equilibriumstrategy, there exist a maximum of two tax rates per µ; tL(µ) and tU(µ); both

24For ¸ > 0; we already excluded that the tax rate could be higher than the second-best taxrate in the preceding subsection.

25

being suboptimal (as compared to the second-best) when they are di¤erent. Moreprecisely, tL(µ) · ´ + ¸µ · tU(µ)). The theorem of the maximum ensures thatthe value of the government’s equilibrium strategy is continuous with respect toµ; therefore functions tL(¢) and tU (¢) are continuous with respect to µ: We denoteby £L and £U the subsets of [µ; µ] leading to a move in the lower and the upperselection respectively. Notice that £L [ £U = [µ; µ] but £L \ £U 6= ; if mixedstrategies are used. For …xing ideas, the following reasoning assumes that ¸ > 0:

The …rst step is to prove that £U is not dense in any interval of [µ; µ]. Wereason by contradiction: take J an interval in [µ; µ] in which £U is dense. Takeµ0 2 J; and a strictly monotonic sequence (µn)n¸1 in £U converging to µ0: Weprove that for all sequences (µn)n¸1; limn!1

tn¡t0µn¡µ0 =

´¡(1+¸)t0t0¡´+¸µ0 ; where tn denotes

tU(µn): Indeed, incentive constraints (µn should not mimic µ0; and vice-versa)imply that:

¡ log(µn + tn)¡´ + µn ¡ ¸tnµn + tn

¸ ¡ log(µ0 + t0)¡´ + µn ¡ ¸t0µ0 + t0

(32)

¡ log(µn + tn)¡´ + µ0 ¡ ¸tnµn + tn

· ¡ log(µ0 + t0)¡´ + µ0 ¡ ¸t0µ0 + t0

(33)

Therefore, taking a …rst-order approximation, and multiplying by (µ0+t0)2 yields

(34)

0 ¸ ((1 + ¸)t¡ ´)(µn ¡ µ0) + (t¡ ´ ¡ ¸µ)(tn ¡ t0) + o(µn ¡ µ0) + o(tn ¡ t0)(35)

0 · ((1 + ¸)t¡ ´)(µn ¡ µ0) + (t¡ ´ ¡ ¸µ)(tn ¡ t0) + o(µn ¡ µ0) + o(tn ¡ t0)The limit of the rate of variations is the same for all sequences, which impliesthat tU is di¤erentiable at µ0; hence di¤erentiable on interval J .

A solution of the di¤erential equation (29) situated above the second-besttaxes is not incentive compatible at any point, because the second-order conditionis never satis…ed. We can conclude that strategy tU is not-incentive compatible,and that interval J does not exist.

It is now easy to conclude that £L is dense in [µ; µ] : being the complemen-tary set (in an interval) of a set £U which is not dense anywhere, £L is dense.Consequently, tL satis…es the di¤erential equation (29) in a dense subset of [µ; µ],which implies that it does so everywhere. The lower selection is necessarily equalto the unique di¤erentiable equilibrium strategy, since we can apply to tL(¢) thereasoning suited for di¤erentiable equilibria.

It remains now to prove that £U contains a …nite number of points. We takeµ1 and µ2 2 £U (where µ1 6= µ2) with corresponding tax rates t1 and t2: Let usdenote by ti(¢) (i = 1; 2) the solution to (29) with maximal de…nition domainpassing through ti at µi. Note that either t1(¢) and t2(¢) are the same, or oneis systematically above the other, because, according to the Cauchy-LipschitzTheorem, two di¤erent solutions to di¤erential equation (29) never cross.

26

Assume for …xing ideas that t2(¢) is above t1(¢): (a) If the two curves are closeenough to each other, t2(µ1) is de…ned and is larger than t1: Notice that t2(µ1)is closer to the second-best than t1: Our study of the incentives when taxes areabove the second-best shows that solutions to the di¤erential equations minimizewelfare (the second-order condition is violated everywhere): when the type isµ1; t2 with belief µ2 is preferred to t2(µ1) with belief µ1: By transitivity, t2 ispreferable to t1 when the true type is µ2: This is in contradiction with incentives.(b) If there is an in…nite number of types in £U ; we can always exhibit µ1 and µ2which are close enough to each other to apply reasoning (a). We conclude that£U contains a …nite number of points.

27

0.2 0.4 0.6 0.8 1 1.2

0.5

1

1.5

2

2.5

λη+1

SBt

θ̂

t

Figure 1: government indifference curves with λ>0.

θ̂

t

),( 11 −− iit θ

1−iI

2−iτ

),( iit θ

),( 11 ++ iit θ

iI 1+iI

1−iτ iτ 1+iτ

skeleton

2−iVτ

1−iVτ

iVτ

1+iVτ

interval in which the policy is chosenindifferent typeindifference curve of type τi

iI

iτ

iVτ

Figure 2: the skeleton.

0.2 0.4 0.6 0.8 1

0.8

0.9

1

1.1

1.2

1.3

θ̂λ

η+1

SBtt

Figure 3: the fully-revealing equilibrium with λ>0.

0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

θ̂SBt

t

λη+1

Figure 4: the fully-revealing equilibrium with λ<0.